11.2 Areas of Regular Polygons

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11.2 Areas of Regular Polygons

• Geometry
• Mrs. Spitz
• Spring 2006

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Objectives/Assignment

• Find the area of an equilateral triangle.
• Find the area of a regular polygon, such as the
  area of a dodecagon.
• In-class 11.2 Worksheet A
• Assignment pp. 672-673 1-32 all

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Finding the area of an equilateral triangle

• The area of any triangle with base length b and
  height h is given by
• A ½bh. The following formula for equilateral
  triangles however, uses ONLY the side length.

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Theorem 11.3 Area of an equilateral triangle

• The area of an equilateral triangle is one fourth
  the square of the length of the side times
• A ¼ s2

s
s
s
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Ex. 2 Finding the area of an Equilateral
Triangle

• Find the area of an equilateral triangle with 8
  inch sides.

Area of an equilateral Triangle
Substitute values.
Simplify.
Multiply ¼ times 64.
A 16
Simplify.
?Using a calculator, the area is about 27.7
square inches.
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More . . .
F

• The apothem is the height of a triangle between
  the center and two consecutive vertices of the
  polygon.
• As in the activity, you can find the area o any
  regular n-gon by dividing the polygon into
  congruent triangles.

A
H
a
E
G
B
D
C
Hexagon ABCDEF with center G, radius GA, and
apothem GH
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More . . .
F

• A Area of 1 triangle of triangles
• ( ½ apothem side length s) of sides
• ½ apothem of sides side length s
• ½ apothem perimeter of a polygon
• This approach can be used to find the area of any
  regular polygon.

A
H
a
E
G
B
D
C
Hexagon ABCDEF with center G, radius GA, and
apothem GH
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Theorem 11.4 Area of a Regular Polygon

• The area of a regular n-gon with side lengths (s)
  is half the product of the apothem (a) and the
  perimeter (P), so
• A ½ aP, or A ½ a ns.
• NOTE In a regular polygon, the length of each
  side is the same. If this length is (s), and
  there are (n) sides, then the perimeter P of the
  polygon is n s, or P ns

The number of congruent triangles formed will be
the same as the number of sides of the polygon.
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More . . .

• A central angle of a regular polygon is an angle
  whose vertex is the center and whose sides
  contain two consecutive vertices of the polygon.
  You can divide 360 by the number of sides to
  find the measure of each central angle of the
  polygon.
• 360/n central angle

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Ex. 3 Finding the area of a regular polygon

• A regular pentagon is inscribed in a circle with
  radius 1 unit. Find the area of the pentagon.

C
1
B
D
1
A
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Solution

• The apply the formula for the area of a regular
  pentagon, you must find its apothem and
  perimeter.
• The measure of central ?ABC is 360, or
  72.

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Solution

• In isosceles triangle ?ABC, the altitude to base
  AC also bisects ?ABC and side AC. The measure of
  ?DBC, then is 36. In right triangle ?BDC, you
  can use trig ratios to find the lengths of the
  legs.

36
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One side

• Reminder rarely in math do you not use
  something you learned in the past chapters. You
  will learn and apply after this.

cos
sin
tan
B
BD
cos 36
36
AD
You have the hypotenuse, you know the degrees . .
. use cosine
1
BD
cos 36
1
cos 36 BD
D
A
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Which one?

• Reminder rarely in math do you not use
  something you learned in the past chapters. You
  will learn and apply after this.

cos
sin
tan
B
DC
36
sin 36
BC
You have the hypotenuse, you know the degrees . .
. use sine
1
1
DC
sin 36
1
sin 36 DC
C
D
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SO . . .

• So the pentagon has an apothem of a BD cos
  36 and a perimeter of P 5(AC) 5(2 DC) 10
  sin 36. Therefore, the area of the pentagon is
• A ½ aP ½ (cos 36)(10 sin 36) ? 2.38 square
  units.

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Ex. 4 Finding the area of a regular dodecagon

• Pendulums. The enclosure on the floor underneath
  the Foucault Pendulum at the Houston Museum of
  Natural Sciences in Houston, Texas, is a regular
  dodecagon with side length of about 4.3 feet and
  a radius of about 8.3 feet. What is the floor
  area of the enclosure?

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Solution

• A dodecagon has 12 sides. So, the perimeter of
  the enclosure is
• P 12(4.3) 51.6 feet

S
8.3 ft.
A
B
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Solution

• In ?SBT, BT ½ (BA) ½ (4.3) 2.15 feet. Use
  the Pythagorean Theorem to find the apothem ST.

a
a ? 8 feet
So, the floor area of the enclosure is
A ½ aP ? ½ (8)(51.6) 206.4 ft. 2
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Upcoming

• I will check Chapter 11 definitions and
  postulates through Thursday COB.
• Notes for 11.2 are only good Thursday for a
  grade.
• Quiz after 11.3. There is no other quiz this
  chapter.